Minimalism and the generalisation problem: on Horwich’s ... · Keywords Truth ·Minimalism ·Generalisation problem 1HorwichianMTand the generalisation problem...

Synthese (2018) 195:1077–1101https://doi.org/10.1007/s11229-016-1227-5

S.I . : MINIMALISM ABOUT TRUTH

Minimalism and the generalisation problem:on Horwich’s second solution

Cezary Cieslinski1

Received: 9 January 2016 / Accepted: 18 September 2016 / Published online: 28 September 2016© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract Disquotational theories of truth are often criticised for being too weak toprove interesting generalisations about truth. In this paper we will propose a certainformal theory to serve as a framework for a solution of the generalisation problem. Incontrast with Horwich’s original proposal, our framework will eschew psychologicalnotions altogether, replacing them with the epistemic notion of believability. The aimwill be to explain why someone who accepts a given disquotational truth theory Th,should also accept various generalisations not provable in Th. The strategywill consistof the development of an axiomatic theory of believability, one permitting us to showhow to derive the believability of generalisations from basic axioms that characterisethe believability predicate, together with the information that Th is a theory of truththat we accept.

Keywords Truth · Minimalism · Generalisation problem

1 Horwichian MT and the generalisation problem

According to Horwich’s minimalism (see Horwich 1999), all the facts about truth canbe explained on the basis of the so-called ‘minimal theory’ (MT ). The axioms of thistheory are the instances of the following disquotational T-schema:

(T) < p > is true iff p

where the expression ‘< p >’ reads ‘the proposition that p’. Horwich claims thatthe minimal theory fully characterises the content of the notion of truth. Moreover,

B Cezary [email protected]

1 Institute of Philosophy, University of Warsaw, Warsaw, Poland

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our understanding of this notion consists of our disposition to accept every (non-paradoxical) instance of (T). The final upshot is that the concept of truth becomeslight and unproblematic, devoid of any deep nature for philosophers then to uncover.

One of the main concerns for the adherent of Horwichian minimalism is the(so-called) generalisation problem. How can the minimalist account for generalitiesinvolving the notion of truth? Consider, for example, the following statements:

(1) Every proposition of the form ‘ϕ → ϕ’ is true;(2) For every ϕ, the negation of ϕ is true iff ϕ is not true;(3) Every theorem of S is true (where S is some theory which we accept).

Objections have been made that Horwich’s minimal theory is too weak to provesuch generalisations (cf. Gupta 1993). The validity of this charge is not entirely clear,with the main stumbling block being that Horwich has never precisely delineated thecollection of axioms of MT . Hence it is not possible to give an exact assessment oftheir truth-theoretic strength. Nevertheless, it is instructive in this context to see howeffective disquotational theories can be in proving general statements of the envisagedsort. On the one hand, it is a well-known fact that some disquotational theories areweak in this respect. As an illustration, let LT be the language obtained by extendingLPA (the language of Peano arithmetic) with a new one-place predicate ‘T (x)’. Theexpressions ‘SentLPA ’ and ‘SentLT ’ will be used to denote sentences of (respectively)LPA and LT . Let I nd(LT ) be the set of all substitutions of the schema of induction‘(ϕ(0) ∧ ∀x [ϕ(x) → ϕ(x + 1)]) → ∀xϕ(x)’ by formulas of LT . We define:

Definition 1

• T B = PA ∪ {T (�ϕ�) ≡ ϕ : ϕ ∈ SentLPA } ∪ I nd(LT ),• UT B = PA ∪ {∀x1 . . . xn[T (�ϕ(x1 . . . xn)�) ≡ ϕ(x1 . . . xn)] : ϕ ∈ LPA} ∪

I nd(LT ),• Theories like T B and UT B but with arithmetical induction only will be denoted(respectively) as T B− and UT B−.

UT B (hence also T B) is a disquotational truth theory quite weak in proving truth-theoretic generalisations. This is the content of the following theorem.

Theorem 2 For every arithmetical formula ϕ(x), if UT B � ∀x[ϕ(x) → T (x)], thenthere is a natural number n such that P A � ∀x[ϕ(x) → Trn(x)].1

It immediately follows that (1) is not derivable in UT B, being that formulas of theform ‘ϕ → ϕ’ have an arbitrarily large complexity. It also follows that for no theoryS, UT B � ∀ψ[PrS(ψ) → T (ψ)] (with ‘PrS(ψ)’ being an arithmetical formulawith the natural reading ‘ψ is provable in S’). The reason here is the same as before;namely, that the syntactical complexity of theorems of S will be arbitrarily large. A

1 For the proof, see Halbach (2001, p. 1960). The expression ‘Trn(x)’ is an arithmetical truth predicate forformulas of complexity not larger than n. The notion of complexity of a formula can be defined in variousways (for example, it can be characterised as the height of the syntactic tree of a formula). We omit thedetails, as they are not crucial here.

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different argument shows that (2)—the compositional principle for negation—is notprovable in UT B either.2

On the other hand, it is not the case that all disquotational theories are truth-theoretically weak. As soon as we drop the typing restrictions, the situation changesdrastically, as witnessed by the following observation due to McGee (1992).

Theorem 3 Let P AT be Peano arithmetic formulated in LT . Let ϕ be an arbitrarysentence of LT . Then there is a sentenceψ of LT such that P AT � ϕ ≡ (T (ψ) ≡ ψ).

In other words, every LT sentence is provably (in PAT ) equivalent to some sub-stitution of Tarski’s disquotational schema. In particular, this includes sentences ofLT corresponding to (1)–(3). Therefore (1)–(3) will be provable in some untypeddisquotational truth theories.

At best, McGee’s result shows that disquotational theories are not doomed at thestart: it is theoretically possible for some set of well-motivated disquotational axiomsto be truth-theoretically strong. However, it still remains unclear whether it is anythingmore than a mere theoretical possibility. The crucial question remains: Is there anyone set of disquotational axioms which is both well-motivated and truth-theoreticallystrong? It is worth observing that the untyped disquotational theories which wereactually proposed in the literature (with some philosophical motivation offered) donot fare well in this respect.3

In effect, the minimalist owes us an answer to the question of why—if at all—weare entitled or perhaps obliged to accept various generalisations involving the notionof truth. Does his disquotational truth theory prove generalisations such as (1)–(3)?And, if it does not, how can it help us to arrive at them? In a nutshell, this is thegeneralisation problem.

A useful way of framing the challenge has been suggested by Ketland (2005),who introduced the concept of conditional epistemic obligation. Ketland starts withthe intuition that if we accept some base arithmetical theory S (formulated in LPA),then we are obliged to accept various further statements, possibly unprovable in Sitself. Here Ketland’s emphasis is on reflection principles; the relevant definition isintroduced below. The acronyms (GR), (UR) and (LR) stand for global, uniform andlocal reflection respectively.

Definition 4

(GR) ∀ψ ∈ LPA [PrS(ψ) → T (ψ)].(UR) ∀x [PrS(ψ(x))) → ψ(x)], for all ψ(x) ∈ LPA.(LR) PrS(ψ) → ψ , for all ψ ∈ SentLPA .

2 The simplest proof known to me uses compactness: given a finite subset Z of axioms ofUT B, a model ofZ can be built which does not satisfy the compositional principle for negation. Hence, adding the negationof (2) to UT B produces a consistent theory.3 PT B and PUT B are examples of such untyped disquotational theories. Their axioms contain substitu-tions of disquotational schemas (the local or the uniform) by positive formulas—formulas of LT in whichevery occurrence of ‘T ’ lies in the scope of even number of negations. However, it is known that none ofthem proves compositional principles for truth. For more information about these theories, see Halbach(2009) and Cieslinski (2011).

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According to Ketland, when accepting S, we are epistemically obliged to acceptreflection principles for S in all three versions. Since none of the reflection principlesis provable in S,4 the natural explanation of our conditional epistemic obligation sug-gests itself: namely, that all three principles become theorems as soon as appropriatetruth axioms are added. However, the prospects for finding well-motivated disquota-tional axioms producing this effect look dim. Therefore, the disquotationalist faces thedilemma. Paraphrasing Ketland, either he strengthens his axioms, thereby rejectingdisquotationalism, or he offers some non-truth-theoretic analysis of the conditionalepistemic obligation.5

Indeed, it is my opinion that Ketland’s concept of conditional epistemic obligationcan be fruitfully applied not just to reflection principles, but also to the compositionalprinciples governing the behaviour of the truth predicate. Consider T B as a startingpoint. For every arithmetical sentenceψ , we can easily establish in T B that T (¬ψ) ≡¬T (ψ). The recognition of this fact carries a conditional epistemic obligation: giventhat we accept T B, we should also accept the general compositional principle fornegation (restricted, in this context, to arithmetical sentences only), even though—asit happens—it is not provable in T B itself. How can the disquotationalist account forthis without compromising his philosophical standpoint? That is the question.6

In recent years Paul Horwich has made two attempts to deal with the challenge.They will be presented and briefly discussed in the next section.

2 Horwich’s two solutions

First attempt In Horwich (1999) an attempt is made to strengthen the minimal theoryin such a way that it proves by itself the desired generalisations. This strengtheninginvolves modifying the proof techniques available to us in MT . In Horwich’s words:

It is plausible [. . .] that there is a truth-preserving rule of inference that can takeus from set of premises attributing to each proposition some property F, to theconclusion that all propositions have F. (Horwich 1999, p. 137)

4 Both (UR) and (LR) permit us to prove the consistency of S and, as such, are unprovable in S by Gödel’ssecond incompleteness theorem. (GR) is formulated in LT , not in LPA , but even if we add the truth predicateand extend S with some natural disquotational truth axioms, the chances are high that (GR) will remainunprovable.5 Cf. Ketland (2005, p. 80). Although Ketland’s discussion concerns conservative truth theories, in myopinion his remarks apply just as well to disquotational theories of truth in general, not necessarilyconservative ones.6 Admittedly, in other places Ketland formulates his criticism in a different manner. Thus, arguing againstTennant, he writes: “Part of the point of the articles by Feferman, Shapiro and myself was to show how toprove reflection principles […] As far as I can see, in the absence of the sort of truth-theoretic justificationgiven by Feferman, Shapiro and myself, Tennant’s proposal is that the deflationist may assume theseprinciples without argument.” (Ketland 2005, p. 85) Here the emphasis is on justifying the independentsentences (namely, reflection principles), not on explaining of why they should be accepted. In this paperI am not going to consider this quite different version of the anti-deflationary argument. Let me say onlythat I do not consider it successful, mainly due to the serious doubts concerning the justificatory value oftruth-theoretic proofs. See Cieslinski (2015, p. 81ff) for more in this direction.

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It seems that Horwich proposes the introduction of a new rule, very similar to thewell-known ω-rule applied in arithmetical contexts. Assuming that we accept all thesentences obtained from ϕ(x) by substituting an arbitrary numeral for the variable x ,the ω-rule permits us to accept the general statement ‘∀xϕ(x)’. The quoted passagehints at a similar strategy. If for each proposition ϕ, F(ϕ) can be derived in our theory,then we are entitled to conclude that ∀ϕF(ϕ).

I will not discuss this idea in detail, referring the reader to Raatikainen (2005) for aconvincing criticism. Themain thrust of Raatikeinen’s remarks is that any rule invokedto solve the generalisation problem should be practical. In other words, there does notseem to be much point in generalisations being provable in a given theory of truth ifwe, as human beings, are never able to produce such proofs. After all, we somehow doreach generalisations about truth and any adequate explanatory account should takethis fact into consideration. Unfortunately, the system with the ω-rule does not satisfythis basic feasibility condition. The rule in question requires infinitely many premisesand for this reason its practical utility is close to null.7 Indeed, I am inclined to thinkthat it is a very serious worry.

Second attempt Horwich’s second solutionwas originally proposed inHorwich (2001)and elaborated on in Horwich (2010). Unlike in the previous case, the current proposalinvolves leaving the proof machinery of MT intact (it remains thoroughly classical);the idea is just to use it together with a certain additional premise. Horwich emphasisesthat, apart fromMT , theminimalist is permitted to uses additional ‘truth-free’ assump-tions in his explanations. For example, we can explain why we accept ‘<Elephantshave trunks> is true’ as soon as we enlarge MT with a truth-free assumption ‘Ele-phants have trunks’. Here is the final answer: we accept that ‘Elephants have trunks’is true because we believe that elephants have trunks and we accept an appropriatedisquotational axiom of MT .

In an attempt to generalise this strategy, Horwich proposes the following truth-freeassumption:

(A) Whenever someone is disposed to accept, for any proposition of structural typeF, that it is G (and to do so for uniform reasons) then he will be disposed to acceptthat every F-proposition is G. (Horwich 2010, p. 45)

With this assumption at hand, Horwich promises to explain why we are inclined toaccept generalisations of the (1)–(3) type. As an example, we present the Horwichianexplanation below for (1).

Explanation 5

(P1) For every proposition of structural type ‘ϕ → ϕ’, we are disposed to accept thatit is true (and we do it for uniform reasons).

(P2) If P1, then we will be disposed to accept that every proposition of structural type‘ϕ → ϕ’ is true.

7 Cf. Raatikainen (2005, p. 176): “Theω-rule has its uses in theoretical contexts, but because of its infinitarynature, it is not a rule of inference in the ordinary sense. That is, the usual rules of inference are decidablerelations between (conclusion) formulas and finite sets of (premiss) formulas. This is not so with the ω-rule.It requires that one can, so to say, have in mind and check infinitely many premisses, and then draw aconclusion. Consequently, we finite human beings are never in a position to apply the ω-rule”.

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Conclusion We will be disposed to accept that every proposition of structural type‘ϕ → ϕ’ is true.

Clearly, the reasoning is logically valid. Indeed, we can assume that premise (P1)describes us as users of a given disquotational theory of truth; it is also easy to observethat P2 is an instance of (A). In effect, we obtain the explanation of our acceptance of‘∀ϕ T (ϕ → ϕ)’.

Still, some critics remained unconvinced. In particular, Bradley Armour-Garb wasdissatisfied with premise P2. In his own words:

One will not be disposed to accept (the proposition) that all F-propositions areG, from the fact that, for any F-proposition, she is disposed to accept that it is G[. . .], unless she is aware of the fact that, for any F-proposition, she is disposedto accept that it is G. (Armour-Garb 2010, p. 699)

Armour-Garb’s reservation seems fair indeed. However, as he notes himself, it ispossible to take this objection into account, which generates the following modifiedversion of Explanation 5:

Explanation 6

S1 For every proposition of structural type ‘ϕ → ϕ’, we are disposed to accept thatit is true.

S2 We are aware that S1.S3 If S1 and S2, then we will be disposed to accept that every proposition of structural

type ‘ϕ → ϕ’ is true.

Conclusion We will be disposed to accept that every proposition of structural type‘ϕ → ϕ’ is true.

Nevertheless, Armour-Garb is dissatisfied with S2. He asks: “What is it for one tobe aware of such a fact”? He then answers:

Here is a plausible answer: for one to be aware of the fact that, for every F-proposition, she is disposed to accept that it is true is for that person to beaware of the fact that she is disposed to accept that every F-proposition is true.(Armour-Garb 2010, p. 700)

If this is so, then S2 simply means ‘we are aware that the conclusion holds’ and forthis reason Armour-Garb accuses Explanation 6 of being viciously circular. We justcannot explain our disposition to accept a general sentence by citing our awarenessthat we have such a disposition.

Still, such a dismissal of Horwichian explanations seems to me too hasty, sincethere are other possible interpretations of S2 which should be taken into account. Thenext section contains an initial sketch of what seems to me to be a more promisingstrategy.

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3 Horwichian explanations reconsidered

Wewill propose here a certain reconstruction of Horwichian explanations. To enhanceclarity, they will be presented in a very restricted, arithmetical framework. For starters,we are going to assume that the disquotational theory T B− is our preferred theory oftruth for the language of arithmetic. The obvious question then arises of why we areinclined to accept such general statements as (1).8 A Horwichian explanation of ouracceptance of ‘∀ψ ∈ SentLPA T (ψ → ψ)’ will be presented below. The explanationis carried out in a metatheory S about which we will make the following stipulations.

(a) The language of S contains expressions ‘we are aware that …’ and ‘we are dis-posed to accept …’, predicated of sentences of LS .

(b) S contains Peano arithmetic.(c) S contains the information that T B− is our theory. That is, we are disposed to

accept sentences in awareness that they are theorems of T B−.(d) S contains an axiom stating that if we are aware that (for every x , ϕ(x)), then for

every x , we are aware that ϕ(x).9

(e) S contains the following necessitation rule: given ϕ as a theorem, we are allowedto infer ‘we are aware that ϕ’.

(f) S contains Horwich’s rule: given a proof of ‘we are aware that for every x , weare disposed to accept ϕ(x)’, we are allowed to infer: ‘we are disposed to accept∀xϕ(x)’.10

It should be noted that, as it stands, S describes us as highly idealised users ofT B−.11 Thus, for example, condition (b) together with the closure condition (e) guar-antees that for every theorem ϕ of PA, S will prove ‘we are aware that ϕ’. Surely, theawareness of every single theorem of Peano arithmetic (including those never provedby anyone) is an impossibly tall order for any real-world agent, which invites thecharge that any solution to the generalisation problem based on S will be as unrealisticand impractical as the recourse to ω-rule. Nonetheless, in fact the situation is not thatdire at all. In the course of explaining the dispositions of the real-world agents, wecan still appeal to those concrete reasonings carried out in S, which employ only the

8 Restricting our attention to Peano arithmetic and T B− brings both gains and losses. On the one hand, wegain clarity, since the set of axioms of T B− (unlike that of MT ) is precisely defined. On the other hand, weadmittedly lose the breadth and scope of Horwich’s original proposal. Indeed, Horwich discusses arbitrarypropositions, not arithmetical sentences, and properties, not formulas. Here we are going to sacrifice scopefor the sake of clarity.However, it isworth emphasising that ifHorwichian explanations donotwork in simplearithmetical contexts, then they are even more problematic when applied to propositions and properties.9 One should be careful, nonetheless, about the use of implication. Let us abbreviate ‘I am aware that x’by ‘A(x)’. Given ‘A

(∀x[ϕ(x) → ψ(x)])’, by stipulation (d) I can infer ‘for every x , A(ϕ(x) → ψ(x)

)’.

‘What I cannot do is to automatically infer ‘for every x , if ϕ(x), then A(ψ(x))’. Even taking this reservationinto account, one could wonder what psychological reality corresponds to (d). My suggestion is that, onthe assumption of a minimal logical competence of the agent, the awareness of the general fact generatessomething more than just a disposition to accept all the instances, namely, the explicit knowledge of asimple algorithm producing, for an arbitrary n, a derivation of ϕ(n) from the general statement.10 This clearly corresponds to Horwich’s assumption (A), even though I formulate it as an inference rulehere.11 I am grateful to the anonymous referee for this observation.

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principles known to the agents at a given time (ideally, principles for which the agentsthemselves have provided proofs).12

At this point let us recall Armour-Garb’s question. What is it for one to be awarethat ϕ? In my opinion, the application of (e) to the real-world agents (see the previousparagraph) provides a reasonable sufficient condition. In order to be aware that ϕ it isenough to prove ϕ. After proving ϕ in our metatheory S, we are permitted to conclude‘we are aware that ϕ’.

Below, we present a Horwich-style explanation of why we are inclined to acceptthat every arithmetical sentence of the form ‘ϕ → ϕ’ is true. The explanation proceedsas follows.

Explanation 7

(1) For every ϕ ∈ SentLT , if we are aware that T B− � ϕ, then we are disposed toaccept ϕ. (By (c))

(2) For every x, T B− � SentLPA (x) → T (x → x). (By (b), since (2) is provablealready in PA)13

(3) We are aware that (2). (Necessitation, applied to (2))(4) For every x, we are aware that T B− � SentLPA (x) → T (x → x). (From (3)

by (d))(5) For every x, SentLT (�SentLPA (x) → T (x → x)�)14 (Provable in PA)(6) For every x, we are disposed to accept: SentLPA (x) → T (x → x). (From (1), (4)

and (5), logic)(7) We are aware that (6). (Necessitation, applied to (6))(8) We are disposed to accept: for every x, [SentLPA (x) → T (x → x)]. (From (7) by

Horwich’s rule)

The acceptance of various other truth-involving generalisations can be explainedin an analogous manner.

How should we assess Explanation 7? All in all, I am inclined to think of it as of astep in the right direction. Nevertheless, I will formulate two critical remarks below,both of which indicate the need for a still further reformulation and amendment.

Before engaging into the criticism, let us note one peculiar trait of Explanation 7(or of Horwichian explanations in general). The explanation of why we are disposedto accept a given statement could proceed by deriving this statement in a theory whichwe accept. Indeed, this would be the case of T B− with ω rule, where various truth-theoretic generalisations become provable; this is also the case when we try to explainour acceptance of the truth of ‘Elephants have trunks’.15 However, this is not whathappens here. The general statement in question, i.e. ‘∀ϕ ∈ LPAT (ϕ → ϕ)’, is not

12 Alternatively, the rules of S could be modified by relativising them to agents and times.13 As usual in such contexts, the intended meaning is that for every x , T B− proves the result of substitutinga numeral denoting x for a free variable in the relevant formula.14 This notation is used here as a shorthand of: ‘for every x , y, if y = �SentLP A (x) → T (x → x)�,then SentLT (y)’, with ‘y = �SentLP A (x) → T (x → x)�’ abbreviating ‘y is the result of substituting anumeral denoting x for a variable v in the expression ‘SentLP A (v) → T (v → v)”.15 ‘T (Elephants have trunks)’ is simply provable in a disquotational truth theory enrichedwith an additionalassumption ‘Elephants have trunks’.

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derived here at all, neither in T B− (which would be impossible, anyway), nor evenin T B− supplemented with some additional premises. What is instead derived is astatement about our disposition to accept the general sentence under discussion.

I will now formulate two objections against Explanation 7 and against Horwichianexplanations in general.

Problem 1 Horwichian explanations are psychological. A psychological fact (namely,our disposition to accept a given sentence) is explained here in terms of our other dis-positions and mental abilities. This in itself is not problematic. There is nothing wrongwith psychological explanations as such. However, the trouble is that in Explanation 7the normative element is completely lost and the following additional question arises:Is someone who accepts T B− (or Horwich’s MT ) committed to accept additionalgeneralisations, unprovable in T B−? Assume for the sake of argument that we dosatisfy the description from Explanation 7, entailing that, while accepting T B−, weare also inclined to accept the general statement ‘∀ϕ ∈ LPAT (ϕ → ϕ)’. But is thereany reason why we should accept such independent sentences?

If the sentence in question was provable in a theory accepted by us, the difficultywould not be so acute. However, as we noticed, Explanation 7 does not contain aderivation of ‘∀ϕ ∈ LPAT (ϕ → ϕ)’ in any theory accepted by us. Why then shouldwe accept it?16

Problem 2 Premise (1) requires some reformulation. It seems that an assumption tothe effect that T B− is a theory accepted by us, should be employed in Horwichianexplanations. Butwhat does itmean to accept a theory? The problem is that Premise (1)does not adequately express this content. For illustration, assume that my knowledgeof the theory T B− is very limited—that I do not know much more about T B− apartfrom the fact that it is some theory. In such a case Premise (1) would be vacuouslytrue; nevertheless, we would not say that in this situation I accept T B−.

In view of this, later on we are going to propose an alternative approach, which pre-serves some essential traits of Horwichian explanations, but gets rid of psychologicalconcepts altogether. However, we will start with remarks about the notion of acceptinga theory.

4 Accepting a theory

Given that we accept a theory Th, why should we then accept various generalisationsthat are not provable in T h? This question is the starting point of our investigations; thisis also what we consider to be the basic challenge behind the generalisation problem.In such a formulation, the notion of accepting a theory comes out as crucial. But whatdoes it mean to accept a theory?

16 Admittedly, the conclusion of Explanation 7 is that we are disposed to accept that ∀ϕ ∈ SentLP A T (ϕ →ϕ). Hence, we could point out that the sentence in question does, after all, follow quite trivially from sometheory accepted by us (namely, from the theory containing this very sentence). However, this would be amoot point. The question still remains how we arrived at such a theory and, more importantly, why weshould embrace it.

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For starters, we are going to consider (and reject) a few candidates for the role ofan explication of ‘I accept Th’. Our criterion of assessment of the explications will betwofold. First, we are going to reject those explicationswhich yield clearly implausibleconsequences.17 Second, we will also reject those explications which are too strong(in a sense to be explained below). Here is our initial list of options. ‘I accept Th’could mean that:

(a) For any sentence ϕ, if I believe that ϕ has a proof in Th, then I am ready to acceptϕ.

(b) For any sentence ϕ, if I believed that ϕ has a proof in Th, then I would be readyto accept ϕ.

(c) I accept that all the theorems of Th are true.(d) I accept some truth-free version of the reflection principle for Th (the local or the

uniform one).

(a) is clearly inadequate for reasonswhich have already been indicated (seeProblem2 above). Indeed, if I knownothing about T h, then (a) is vacuously true. Hence it wouldfollow that I accept Th, which is hardly plausible.

(b)—a counterfactual strengthening of (a)—turns out to be inadequate as well. Isubmit that very few people would accept any theory Th in such a sense! For example,if I believed that Peano arithmetic proves ‘0 = 1’, then I would not be ready to acceptthat 0 = 1. I would reject PA instead.

Even though (c) sounds plausible in itself, it is too strong to be of much use inour present discussion. To give a simple example, one of the ‘conditional epistemicobligations’ (to use Ketland’s phrase again) is the consistency of Th. Why is it thatwhen accepting Th, we should accept that Th is consistent? In a way, the explication(c) makes the issue trivial. Thus, given an arithmetical theory Th it is easy to observethat any theory of truth which includes T-sentences for the arithmetical language,proves ConTh (the consistency of Th) when supplemented with ‘All theorems of Thare true’. So far so good—but where does it leave the disquotationalist? How can heaccept Th if—as it may well happen—his disquotational truth theory for the languageof Th does not permit him to prove ‘all theorems of Th are true’? Indeed, if (c) wasthe only possible way to make sense of the notion of ‘accepting a theory’, I would beinclined to see it as a strong argument against disquotationalism. But, this is not theonly way.

In addition, treating (c) as an explication of ‘I accept Th’ will be very problematicin some particularly pertinent cases, namely, when Th itself is a theory of truth.For illustration, let Th be K F + con; in other words, let it be the Kripke–Fefermansystem supplemented with the consistency axiom ‘∀ϕ¬(T (ϕ)∧T (¬ϕ))’.18What does

17 For example, if under a given explication I ‘accept’ a theory which I do not accept in a normal sense ofthe word, this will count heavily against the proposed explication.18 See Reinhardt (1986) and Feferman (1991), where the theory in question was introduced; cf. alsoFeferman (1984). K F was meant to capture a Kripkean notion of truth and originally it was formulatedin the language with two primitive predicates ‘T ’ and ‘F’ for truth and falsity respectively. For the list ofaxioms of K F in the language with ‘T ’ but without the falsity predicate, see Halbach (2011, pp. 200–201).

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it mean to accept such a theory? Since the system K F + con proves the untruth ofsome of its own theorems, accepting K F + con in the sense (c)—that is, introducingthe information that all theorems of K F + con are true—produces an inconsistenttheory.19 This is another reason why (c) should be deemed unsatisfactory.

We reject also (d) for similar reasons as (c)—that is, we consider it too strongfor our purposes. If ‘accepting Th’ means accepting all the substitutions of (say) thelocal reflection principle for Th, then how can the disquotationalist accept Th in thissense if (as it may well happen) his disquotational truth theory does not prove allsuch substitutions? Indeed, one could try to argue that on any admissible sense of‘accepting’, our acceptance of Th commits us to some forms of reflection for Th.20

Still, I am inclined to view such a commitment as something to be explained. It istotally unhelpful to postulate it in advance, as a part of the meaning of ‘I accept Th’.

At this point we have considered and rejected four explications of ‘I accept Th’.So what is left?

In this paper we will work with the following explication, which is a modifiedversion of (b).

(e) For any sentence ϕ, if I believed that ϕ has a proof in Th and I had no independentreason to disbelieve ϕ, then I would be ready to accept ϕ.

Observe that the objection raised earlier against (b) is no longer valid. Here, a merepossibility of the theory being inconsistent is no longer a problem: if I believed that‘0 = 1’ has a proof in Th, then I would not be ready to accept that 0 = 1 becauseof independent reasons! In fact, the formulation (e) gives justice to the fact that werarely—if ever—accept our theories unconditionally. The intuition is rather that wewill stick to them as long as we believe that they do not yield false consequences. Still,if I accept Th, then given a new sentence ϕ (new in the sense that I neither acceptednor rejected ϕ previously), I would accept ϕ if I believed that it is a theorem of Th.Indeed, this is how the present story goes.

Throughout the rest of this paper I am going to treat (e) as my basic description ofthe content of ‘I accept Th’. Let us observe that, with such a choice, the generalisationchallenge remains nontrivial. For example, we still have to explain why someoneaccepting PA in this sense should be committed to accept statements not provable inPA (the consistency statement in particular). Alternatively, one could claim that anysuch additional commitments are illusory.

19 For a sentence L such that K F � L ≡ ¬T (L), it is possible to show that K F + con � L and thereforeK F + con � ¬T (L). It is then easy to observe that K F + con+ ‘All theorems of K F + con are true’proves T (L) and hence it is inconsistent. See Halbach (2011, p. 215).20 Cf. Ketland’s ‘If one accepts a mathematical base theory S, then one is committed to accepting a numberof further statements in the language of the base theory (and one of these is the Gödel sentence G)’. Ketland(2005, p. 79) In this context, the role of reflection principles is quite central: they give us ‘the possibilityof systematically generating larger and larger systems whose acceptability is implicit in acceptance of thestarting theory. The engines for that purpose arewhat have come to be called reflection principles’ (Feferman1991, p. 1). These words of Feferman are quoted with approval by Ketland in the same paper.

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5 An epistemic approach

5.1 Introducing believability

In the remaining part of this paper, a Horwich-style solution to the generalisationproblem will be proposed, one which eschews psychological concepts altogether.Instead of ‘being aware of’ and ‘being disposed to accept’ (see Explanation 7), in ouramended explanations we are going to employ a single epistemic predicate B(x) (‘xis believable’), predicated of sentences. The intuitive intended interpretation of B(x)is ‘there is a reason to accept x which is good enough to warrant rational acceptance ofx , given the absence of reasons to reject x’ (later on we will abbreviate this as ‘there isa reason to accept x which is normally good enough’ or just as ‘there is a good reasonto accept x’). We emphasise that this interpretation is not to be confused with ‘thereis a compelling reason to accept x’ or with an unconditional ‘x should be rationallyaccepted’. It corresponds rather to the weak notion of theory acceptance characterisedby condition (e) on p. 11 of this paper. The initial idea is that proofs carried out in atheory that we accept are treated by us exactly as such reasons: when presented withsuch a proof, we accept its conclusion, unless given strong reasons to the contrary.21

In a moment we are going to characterise the new predicate by means of axioms andrules. However, I will start with a general outline of the proposed strategy.

We begin with a very simple notion of truth, characterised by purely disquotationalaxioms of our truth theory Th (resembling T B− or, more ambitiously, resemblingperhaps Horwich’s MT ). We are convinced that our axioms fully specify the meaningof the truth predicate. Moreover, we treat the axioms as obvious, simple and episte-mologically basic. Sometimes we express these convictions by slogans like ‘truth isinnocent’ or ‘truth is a light notion’.

We accept our disquotational truth theory. The notion of acceptance is employedhere in the sense (e) of the previous section. In practice, given a reliable informationabout the existence of a proof of ϕ in Th, we accept ϕ. However, there is one additionalelement of the picture: it should be emphasised that in such cases we accept ϕ becauseof the proof in Th. Indeed, we consider theorems of Th believable. The key point isthat our mathematical practice—that of accepting statements because of their proofsin Th (even in cases when we did not check the proofs by ourselves)—would beirrational without the underlying belief that proofs in Th function as reasons whichare normally good enough to accept their conclusions.

In the next stage, we characterise our notion of believability bymeans of some basicaxioms and rules. As we are going to see, this move brings important consequences.The final result is that we declare as believable various additional statements in thelanguage of Th, unprovable in Th itself. In effect, our initial acceptance of disquota-tional theory Th, together with some basic convictions about believability, leads us to

21 Similarly, information froma very trustworthywitnesswill warrant rational acceptance given the absenceof reasons to reject the testimony (in particular, given the absence of contrary testimonies from othertrustworthy witnesses). On the other hand, information conveyed by an unreliable witness is not believablein our intended sense: the reason in question (namely, the witness’s words) does not warrant rationalacceptance even in the absence of any evidence to the contrary—this is the intuition.

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the recognition that, indeed, there is a reason to accept e.g. compositional truth axioms.In this way new commitments are generated. The point is that given such a reason,we behave rationally when—not aware of any good reasons to reject compositionalprinciples—we finally accept them. This is the general outline of the story proposedhere.

In order to make some of these ideas precise, we introduce the following definition:

Definition 8 Let K be an axiomatisable extension of PA in the language LK (possiblyricher than LPA). Denote as LK ,B the extension of LK with a new one-place predicate‘B’. Let K B be a theory K formulated in the language LK ,B .22

• We denote as Bel(K )− the theory in the language LK ,B which extends K B withthe following axioms:

(A1)∀ψ ∈ LK ,B[K B � ψ → B(ψ)](A2)∀ϕ,ψ ∈ LK ,B[(B(ϕ) ∧ B(ϕ → ψ)

) → B(ψ)]In addition, the theory Bel(K )− has the following rules of inference:

nec� φ

� B(φ)

� ∀x Bφ(x)

� B(∀xφ(x)

) gen

• We denote as BelCon(K )− the theory which is exactly like Bel(K )−, except thatit contains the following additional consistency axiom:

(A3)∀ψ ∈ LK ,B¬B(ψ ∧ ¬ψ).

• Bel(K ) and BelCon(K ) are theories which are exactly like Bel(K )− andBelCon(K )−, except that they contain all the axioms of induction for formulasof LK ,B .

We will view Bel(K ) as a believability theory built over our initial theory K . Theintended reading of ‘B(x)’ (‘x is believable’) will manifest itself in our treatment of aproof of B(ψ) in Bel(K ) as showing that ψ should be rationally accepted, given ouracceptance of K and given that we are not aware of any independent reasons to rejectψ .23

Axiom (A1) is not muchmore than a formalisation of the assumption hidden behindour acceptance of K . We realise that proofs in K are (for us) good reasons to accepttheir conclusions—that is the content. The only addition is that (A1) also expressesour acceptance of logic in the extended language, with the new predicate ‘B’ (weemphasise that in the formulation of (A1) the provability predicate for K B is used).

Axiom (A2) expresses the following thought: if there is a good reason to accept theimplication and there is a good reason to accept its antecedent, then there is a goodreason to accept the consequent.

As for the rules of inference of Bel(K ), the intuitive validity of nec seems tome uncontroversial. If a proof of ϕ in Bel(K ) is provided, it is simply this proof

22 The only difference between K and K B is that K B contains logical axioms also in the language withthe new predicate.23 As noted earlier, the last reservation is clearly forced by our weak notion (e) of accepting a theory.

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itself which constitutes a good reason to accept ϕ, therefore in such a situation ϕ isbelievable. 24

Rule gen is of crucial importance. As we are going to see, it is exactly gen whichpermits us to derive (in the scope of ‘B’) strong consequences, unprovable in K itself.In order to explain the idea behind gen, let us compare it to the following (somewhatsimilar) reflection rule for the theory K 25:

� ∀x PrKφ(x)

� ∀xφ(x)ref

Let us assume that K is the theory we accept. Let us also assume that we are able toprove (perhaps in K itself) that for every x , φ(x) is a theorem of K . Why should wethen accept the general statement ‘∀x φ(x)’? The answer becomes unproblematic assoon as we assume that all theorems of K are true. Indeed, after this (with just a fewbasic properties of the truth predicate) the conclusion of ref can be easily justified.Still, with just the disquotational notion of truth we might not have a right to such anassumption. How can we close this gap?26

The idea of gen is to close the gap by introducing an additional epistemologicalelement. Here, it should be noted that what we really do in such contexts involves somereasoning about reasons, not about truth. For illustration, assume that you consider thePope an authority in matters of religious faith. You trust the Pope and the fact that acertain opinion was expressed by the Pope is treated by you as a good reason to acceptthis opinion. Assume in addition that you heard from a trustworthy source—possiblyfrom the Pope himself—that every book written by a given theologian was proclaimedheretical by the Pope. You will then have a good reason to think that (for every bookwritten by the theologian, there is a good reason to think that it contains a heresy).The idea behind gen is that this in itself constitutes a good reason to think that everybook written by the theologian contains a heresy. According to the present proposal,this is how we reason about reasons.

It is worth observing that the following transformation of gen into the axiommissesthe mark:

(Ax) ∀ψ[∀nB(ψ(n)) → B(∀nψ(n))].As a formalisation of our intuitions concerningbelievability, (Ax) is clearly inadequate.For every numerical instance of ψ , there may exist a good reason to accept it withoutthere being a uniform reason, one covering all the cases simultaneously. In otherwords,

24 Note that, unlike in the case of (A1), nec is the closure condition for Bel(K ), so it can be applied alsoto statements which are not theorems of K B. On the other hand, nec by itself has not the full force of (A1).Indeed, for every theorem ϕ of K B, it is enough to have nec (without (A1)) in order to prove B(ϕ). Still,nec by itself is not enough to derive the believability of all theorems of K B as a general statement.25 See Beklemishev (2005) for a more detailed discussion of the reflection rule. In particular, Beklemishevshows that ref is equivalent to (UR) over Peano arithmetic.26 Indeed, one might be tempted to interpret Horwich’s assumption (A) as containing the proposal ofenriching MT with something like ref (or maybe even with the uniform reflection principle stated in theform of an implication). However, the key question would then be what gives us the right to introduce sucha rule—why should we accept its conclusion, given the acceptance of the premise?

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theremight still not be any good reason to believe the general statement in question.Weemphasise that in gen this particular flaw is eliminated; we require that the statement‘∀nB(ψ(n))’ is itself provable, and that we therefore have a good, uniform reason(namely, the proof of the general sentence in Bel(K )) to believe of every instance ofψ that it is believable. The intuition is that such a good reason to think that all instancesare believable is also a good reason to believe a general statement. It is exactly thisintuition that is expressed by gen.

Axiom (A3) is an ugly duckling and I do not present it as a part of the minimalbelievability theory Bel(K ). In this context we should emphasise one more time that,according to the intended interpretation, believability is neither truth nor unconditionalrational acceptance. Not only can false sentences be believable but there is also noautomatic transition from ‘B(ϕ)’ to ‘ϕ should be rationally accepted’. Intuitively,‘B(ϕ)’ instead means that ‘there is a reason to accept ϕ which is normally goodenough’ or, in other words, ‘you should rationally accept ϕ unless you have a strongindependent reason to reject ϕ’. Observe that this reading corresponds closely to thecharacterisation of the weak notion of theory acceptance adopted in this paper. Thus,the information that ϕ is provable in K (a theory which I accept) is normally goodenough for me to accept ϕ. However, my acceptance of K is not unconditional and ifit turned out that K � 0 = 1, then I would not be ready to accept that 0 = 1. In ourframework, this would be the situation of Bel(K ) proving B(0 = 1). I then still havea reason to accept ‘0 = 1’ that is normally good enough (namely, the proof in K—atheory which I accept). However, the situation envisaged is not normal, since I amalso able to prove that B(¬0 = 1). In general, the intuition is that whenever we obtainboth B(ϕ) and B(¬ϕ), there is no automatic transition to the rational acceptance ofstatements in the scope of the believability predicate.

All in all, one might have a good reason—that is, a reason normally good enough—to accept a sentence ϕ (for instance, a derivation of ϕ from some plausible premises)while at the same time having a good reason to accept¬ϕ (say, a derivation of¬ϕ fromanother set of plausible premises). This is why I reject (A3) as an axiom formalisingthe properties of the believability predicate. The axiom is introduced just in order toindicate that nice models (in a sense to be explained) can be provided even for fullBelCon(K ) on the assumption that the initial theory K is ‘safe’ enough. Nevertheless,none of the applications of the believability theory discussed in this paper will employ(A3).

5.2 Formal properties of Bel(K )

In this section we state the main formal results about the properties of Bel(K ). Weomit the proofs, which can be found in the Appendix.

When analysing formal properties of Bel(K ) with the intended interpretation of‘B’ in mind, the following question looms as primary. Imagine that K is an initialtheory which we accept. If K is trustworthy, just how trustworthy are the statementswhich are, provably in Bel(K ), within the scope of B? In other words, if Bel(K )

proves the believability of a given statement, will something go wrong if we acceptthis statement? In order to facilitate further discussion, we introduce the followingnotational convention:

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Definition 9 I ntBel(K ) = {ψ ∈ LK ,B : Bel(K ) � B(ψ)}The set I ntBelCon(K ) can be defined in an analogous manner.

What we would like to know is whether the elements of I ntBel(K ) form a nicetheory. One particular interpretation of the phrase, ‘a nice theory’, will be consideredhere; that is, nice theories are interpretable in the standard model of arithmetic.27 Oneof our main formal observation is that even I ntBelCon(K ) is indeed nice in this sense.This is the content of the theorem formulated below.

Theorem 10 Let K be a theory in the language without the predicate ‘B’. If N (thestandardmodel of arithmetic) is expandable to amodel N∗ of K , then N∗ is expandableto a model of I ntBelCon(K ).

Having checked that I ntBel(K ) is a nice theory, we observe that even without theaxiom (A3), the theory Bel(T B−), obtained by taking T B− as the basic (initiallyaccepted) theory, is already strong enough to prove the believability of various inter-esting generalisations unprovable in T B−. Let ‘B(CT )’ be a shorthand of ‘all thecompositional truth principles are believable’. It is then possible to show that:

Theorem 11 Bel(T B−) � B(CT ).

As a final observation, we remark that similar results can be obtained for someuntyped disquotational truth theories taken as a starting point. Thus, in (Horsten andLeigh 2015) the language LT,F has been defined in the following manner.

Definition 12

• Terms and function symbols of LT,F are exactly those of LPA.• The connectives of LT,F are ∧ and ∨.• The predicates of LT,F are: =, �=, T , F .• In LT,F we have the quantifiers ∀ and ∃.• The formulas of LT,F are built in the usual style.

What is important is that in LT,F we do not have a symbol for negation. We willinstead be using the symmetric relation of being a dual formula. Strictly speaking, thisnotion is first defined for primitive symbols and then generalised to cover all formulas.

Definition 13

• ‘=’ and ‘ �=’ are dual predicate symbols.• ‘T ’ and ‘F’ are dual predicate symbols.• ‘∧’ and ‘∨’ are dual connectives.• ‘∀’ and ‘∃’ are dual quantifiers.• ψ and ϕ are dual formulas iff ψ is obtained from ϕ by replacing every symbol in

ϕ with its dual.

We will use the notation ϕd whenever we want to indicate that a given formula is adual of ϕ.

27 It follows in particular that nice theories are ω-consistent.

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In this language, the following basic disquotational theory can be characterised:

Definition 14

• T FB = PA ∪ {T (�ϕ�) ≡ ϕ : ϕ ∈ LT,F } ∪ {F(�ϕd�) ≡ ϕ : ϕ ∈ LT,F } ∪I nd(LT,F ).

• A theory like T FB, butwith arithmetical induction only,will be denoted as T FB−.

Let K F be a variant of Kripke–Feferman truth theory in the language LT,F .28 Let‘B(K F)’ be the sentence stating that all compositional axioms of K F are believable.Then it is possible to show that:

Theorem 15 Bel(T FB−) � B(K F).

In this way a Horwich-style solution to the generalisation problem is vindicated.Indeed, it turns out that if we find our initial disquotational theory (for example,T B− or T FB−) believable, then we will also find believable various truth-theoreticgeneralisations, not provable in the initial theory of our choice.

5.3 How does it help?

How attractive is the proposed epistemic strategy? Does it really permit the disquota-tionalist to escape the generalisation problem?We believe that it does. Nevertheless, inthis final section we are going to discuss some limitations of the proposed believabilityframework, indicating the areas with an additional work still awaiting to be done.

For starters, let us recapitulate the proposed solution.The disquotationalist begins with accepting some disquotational theory K , which

may well be truth-theoretically weak. In particular, it might happen that various com-positional principles involving truth are not provable in K . Should he accept theseprinciples? If so, why? This is the question.

Here is the proposed answer. The disquotationalist treats proofs in K as goodreasons to accept their conclusions (more exactly, he treats them as reasons which arenormally good enough—see the condition (e) earlier in this paper). In this realisation,he applies the believability theory to K treating it as the base.However, it transpires thatBel(K ) proves the believability of compositional principles. The initial acceptance ofK , together with some basic convictions about believability, leads the disquotationalistto the recognition that there is indeed a good reason to accept compositional principles.Given such a good reason and without being aware of any good reason to reject theseprinciples, he should accept them.

Viewed in more general terms, our endeavour has been to understand the commit-ments of someone who accepts a given theory K .29 We have noticed that the answerdepends on the notion of accepting a theory. After observing that some strong notionstrivialise the issue, we have decided to focus on the weak notion of theory acceptance.This has forced us to adopt a notion of believability partially characterised by theaxioms (A1) and (A2) but without the consistency axiom (A3). We have argued that

28 See (Horsten and Leigh 2015) for the full list of axioms.29 For a classical analysis of such commitments, we refer the reader to Feferman (1991).

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the statement ‘All theorems of K are believable’ should be accepted once we reflectupon our mathematical practice as users of K .30 That is, once we realise that “if webelieved that ϕ has a proof in K andwe had no independent reason to disbelieve ϕ, thenwe would be ready to accept ϕ”; in other words, once we appreciate that proofs in Kare normally good enough for us as reasons to accept their conclusions. The resultingtheory Bel(K ), with sentences declared believable even though they are unprovablein K , permits us then to recognise our commitments as users of K .

At this point it should be emphasised that our intended interpretation of ‘B’ isonly partially captured by the axioms and rules of Bel(K ). Apart from the formalmachinery of the believability theory, the second crucial element of the picture, externalto Bel(K ), are the prospective bridge rules which connect believability with rationalacceptance. When then, or under what conditions, are we permitted to move from‘B(ϕ)’ to ‘ϕ should be rationally accepted’? In this paper I have advocated a modestapproach. I would say that given a proof of B(ϕ) in Bel(K ), we should rationallyaccept ϕ as long as we are not aware of any derivation of B(¬ϕ) in Bel(K ) or insome theory Bel(K ′), built over a well-entrenched mathematical theory K ′, possiblydifferent than K . I consider it to be a workable option, one that the deflationist canemploy in practice, successfully defending himself against the objections of the critics.However, there is still a room for a more ambitious approach, aiming to explain howwe ‘reason about reasons’ even in troublesome cases.

In general,whenworkingwithout the consistency axiom (A3) (as I thinkwe should),we are faced with the question of how the bridge rules are to be used in contexts withcontradictory statements appearing within the scope of B. One of the main issuesis the problem of logical explosion. Since by (A1) the whole of first-order logic isbelievable, it is easy to see that with contradictory statements in the scope of B, anarbitrary statement will become believable, provably so in our theory. Let me stressthat with our weak reading of ‘B’, this result is not troublesome in itself. Indeed, theintuition is then that treating proofs in K as reasons normally good enough to rationallyaccept their conclusions, gives you a reason to accept an arbitrary ϕ—again, a reasonwhich is normally good enough to warrant rational acceptance. However, with bothB(ϕ) and B(¬ϕ) as theorems, the situation is not normal and the bridge rules shouldnot licence the universal transition from B(x) to ‘x should be rationally accepted’.Indeed, various concrete cases indicate that the problem concerns not so much thebelievability rules and axioms, but K itself.31

30 This move would not be valid with the consistency axiom (A3) treated as a part of our characterisationof the notion of believability. The problem is that I just cannot see why ‘all theorems of K are believable’should be then rationally accepted, given that our initial notion of accepting a theory is so weak. What isit that entitles us to conclude immediately that proofs in K will always function as compelling reasons toaccept their conclusions? What is the guarantee that nothing will go wrong—that no independent reason toreject a theorem of K will ever be provided? I cannot see a good answer to this question.31 An example is provided by the truth theory FS, known to be ω-inconsistent (see Halbach 2011, pp.157–158). As it happens, the proof of ω-inconsistency of FS can be reconstructed in PA; in other words,there is a formula ϕ(x) ∈ LT such that PA proves that: (1) FS � ∃xϕ(x), (2) ∀x FS � ¬ϕ(x). Then axiom(A1) of Bel(FS) permits us to conclude that B

(∃xϕ(x))and ∀x B(¬ϕ(x)

), which in turn (by gen) gives us

B(∀x¬ϕ(x)

). In this way a contradiction in the scope of B is obtained, which only reflects the troublesome

properties of FS itself.

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Nevertheless, in themore ambitious approach afore-mentioned, one could try to givejustice to the fact that we domake great efforts to tolerate contradictions in the scope ofB. Naturally, when given good reasons for a pair of contradictory statements, we try toresolve the contradiction and to reassess the reasons. However, themain observation tomakewould be that, in themeantime,we do not treat the believability of a contradictionas an argument which, for every sentence ϕ of our language, demonstrates that rationalacceptance of ϕ is invalid. As I have said, one bridge rule will permit us to concludethat ϕ should be rationally accepted, given that we have B(ϕ) and we are not awareof any reason to accept the negation of ϕ. However, blocking the move to ‘we shouldrationally accept ϕ’ whenever both B(ϕ) and B(¬ϕ) are provable, always and withoutexceptions, does not seem to be a good idea because in practice such an additional ruleis not treated by us as universal. This intuition could lead to a more general, possiblyparaconsistent analysis of both believability and rational acceptance. Indeed, I viewit as a promising line of research. Nevertheless, here I prefer to remain noncommittalabout the shape of such a general theory of believability and rational acceptance.

An additional avenue for further research corresponds to the limiting conditionbuilt into our Theorem 10: namely, to the reservation that the base theory K is tobe formulated in a language without ‘B’. Indeed, it is easy to observe that removingthis reservation makes Theorem 10 false. For a trivial example, let K be PA withthe additional axiom ‘B(0 = 1)’. Then clearly the standard model of arithmetic isexpandable to a model of K but just as clearly I ntBel(K ) is inconsistent. A non-trivialillustration is provided by the lottery paradox, introduced by Kyburg (1961). Considera fair lottery with a large number of tickets exactly one of which will win. In whatfollows we assume that the tickets are numbered from 1 to k. It is highly probable thatticket 1 will not win, ticket 2 will not win …and so on, separately for every singleticket up to k.

The following intuitively plausible premises then lead to a contradiction:

1. It is rational to accept propositions with very high probability assigned to them.2. If you are aware that a given proposition is inconsistent, then it is not rational to

accept it.3. If it is rational to accept ϕ and it is rational to accept ψ , then it is rational to accept

ϕ ∧ ψ .

Let R(ϕ) abbreviate ‘it is rational to accept ϕ’. We obtain:

(a) R(ticket 1 will not win) and R(ticket 2 will not win) and …R(ticket k will notwin), (by 1)

(b) R(ticket 1 will not win and ticket 2 will not win and …ticket k will not win),(by 3)

(c) R(ticket 1 will win or …ticket k will win), (by the assumption that some ticketwill win)

(d) R((ticket 1 will not win and …ticket k will not win) and (ticket 1 will win or

…ticket k will win))(by 3)

However, we are aware that the last sentence in the scope of R is inconsistent andin this way a contradiction with 2 is obtained.

It should be emphasised that the lottery paradox concerns rational acceptance, notbelievability. For believability, premise 2 simply does not hold. Believability of ϕ

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and ψ (separately) means, in intuitive terms, that we should rationally accept both ϕ

and ψ assuming that we are not aware of any good reason to accept ¬ϕ or ¬ψ . Weshould then rationally accept (ϕ ∧ ψ), unless we are aware of a good reason to accept¬(ϕ ∧ ψ), with the point being simply that a good justification of both conjuncts isnormally good enough to warrant rational acceptance of a conjunction.32 However, ifwe are aware that ¬(ϕ ∧ψ) is a truth of logic, then the situation is not normal and theusual bridge rule leading to rational acceptance does not apply.

Nevertheless, steps (a)-(c) could be reproduced in Bel(K ) assuming that the back-ground theory K , formulated in the language with ‘B’, contains as its theorems (i)all sentences ‘B(ticket m will not win)’ for 1 ≤ m ≤ k, (ii) ‘B(ticket 1 will win or…ticket k will win)’. Observe that with these assumptions about K , the rule gen isnot needed at all to derive a contradiction in the scope of B, as the mere believabilityof propositional logic together with (A2) is quite enough to produce this effect.

As before, if a contradiction appears in the scope of ‘B’, then the bridge rules hadbetter not block the rational acceptance of all sentences of our language; in otherwords, the transition to rational acceptance should be invalidated locally, not globally.Let me also stress that the present proposal is not meant to offer a solution to the lotteryparadox. Indeed, I think that the various solutions proposed in the literature can beapplied in our framework on the level of the bridge rules, with the believability theoryremaining neutral in this respect.33

In spite of these limitations, I think that the proposed epistemic strategy indeedoffers an attractive way out for the disquotationalist who is worried about the truth-theoretic weakness of his axioms. It permits him to vindicate his philosophical positionin a model case of arithmetical truth, where Theorem 10 applies. There are also goodprospects for moving towards a conception of a more directly self-applicable notionof believability, with a base theory K possibly being formulated in the languagecontaining ‘B’.34

Acknowledgements The research presented in this paper was supported by the National Science Centre,Poland (NCN), grant number 2014/13/B/HS1/02892.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate if changes were made.

32 See Ryan (1996, p. 124) for a forceful exposition of this view. In her words, ‘Apart from the fact that[a conjunction principle], along with the other principles and the lottery story, generates a paradox, thisappears to be an innocent epistemic principle.’ Cf. also Douven (2002, p. 394), where amore general closureprinciple is described as one that ‘we rely on in many of our everyday deliberations’.33 For example, if someone accepts premises 1 and 2 of the reasoning leading to the lottery paradox whileclaiming, with Kyburg, that rational acceptance does not agglomerate (in other words, that you cannot derive‘ϕ ∧ ψ should be rationally accepted’ from ‘ϕ should be rationally accepted’ and ‘ψ should be rationallyaccepted’), the proposed bridge rules would licence the transition to rational acceptance of all statementsof the form ‘ticket m will not win’ while blocking the rational acceptance of their conjunction.34 Apart from introducing bridge rules for handling contradictions, another possible move would consist initerating the application of the believability axioms and rules. For illustration, let K0 be PA; define Kn+1as I ntBelCon(Kn )

. We conjecture that a safety result similar to Theorem 10 can be obtained for all theoriesKn in this sequence.

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Appendix

In the proof of Theorem 10, the following two hierarchies will be used.

Definition 16

• S0 = K B ∪ Axioms (A1)-(A3) of BelCon(K ),• Sn+1 = Sn ∪ {B(ψ) : Sn � ψ} ∪ {∀xψ(x) : Sn � ∀x Bψ(x)},• Sω = ⋃

n∈NSn .

We emphasise that the S-sets will be treated here as theories (that is, as closedunder first-order consequence). Accordingly, the intended reading is that the set (say)S1 contains everything which can be proved from S0 with the help of additionalaxioms of the form ‘B(ψ)’ and ‘∀xψ(x)’, satisfying the appropriate conditions fromDefinition 16.

Definition 17 For a model N∗ of K , we define:

• B0 = K B,• Bn+1 = {ψ : ∀Z ⊇ Bn[ if (N∗, Z) |� (A2) ∧ (A3), then (N∗, Z) |� ψ]},• Bω = ⋃

n∈NBn .

The proof of Theorem 10 consists in showing that: (1) I ntBelCon(K ) ⊆ Sω; (2)(N∗, Bω) |� Sω. Then it follows immediately that a certain expansion of N∗, namely,(N∗, Bω), is a model of I ntBelCon(K ).Below we provide some details, starting with the following observation:

Observation 18 ∀n Bn ⊆ Bn+1.

This can be easily proved by induction. In particular, for n = 0, it is enough toobserve that theorems of K B (that is, elements of B0) are true in all structures (N∗, Z),independently of the choice of Z .35

It is also very easy to verify that:

Observation 19 ∀n (N∗, Bn) |� (A1) − (A3).

The content of the next fact is that all the S-sets are contained in the correspondingsets from the B-hierarchy.

Fact 20 ∀n Sn ⊆ Bn+1.

Proof It is easy to observe that S0 ⊆ B1.36 For the inductive part, assuming thatSn ⊆ Bn+1, we show that Sn+1 ⊆ Bn+2. Fix ψ ∈ Sn+1 and let (α1 . . . αs) be a proofof ψ from the axioms of Sn+1.37 We are going to show that ∀k ≤ s αk ∈ Bn+2.

35 We remind the reader that K B has been defined as K (a theory in the language LK , not containing ‘B’),but formulated in the language LK ,B which is an extension of LK with a new one place predicate ‘B’. Tobe more precise, this means that the only axioms of K B employing the new predicate ‘B(x)’ are logicalaxioms. In effect, an arbitrary interpretation Z of ‘B’ will make them true.36 Trivially, K B ⊆ B1; it is also immediate that (A1)–(A3) remain true in every expansion (N∗, Z) giventhat Z ⊇ B0 and (N∗, Z) |� (A2) ∧ (A3). Finally, the axioms of induction for LK ,B will remain trueindependently of the choice of Z .37 See the remark immediately below Definition 16.

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Fix k ≤ s and assume that ∀i < k αi ∈ Bn+2. We obtain the appropriate conclusionfor αk by considering the following cases.Case 1 αk ∈ Sn—then by the inductive assumption and Observation 18, αk ∈ Bn+2.Case 2 αk = B(ψ) forψ ∈ Sn . Then by the inductive assumptionψ ∈ Bn+1, therefore∀Z ⊇ Bn+1(N∗, Z) |� B(ψ). So B(ψ) ∈ Bn+2.Case 3 αk = ∀xψ(x) and Sn � ∀x Bψ(x). In this case, by the inductive assumption∀x Bψ(x) ∈ Bn+1, therefore:

∀Z ⊇ Bn[(N∗, Z) |� (A2) ∧ (A3) → (N∗, Z) |� ∀x Bψ(x)].In effect:

∀Z ⊇ Bn[(N∗, Z) |� (A2) ∧ (A3) → ∀x (ψ(x) ∈ Z)].By Observation 19, it immediately follows that

∀x (ψ(x) ∈ Bn).

Now we will consider two subcases:(a) n = 0. Then ∀x (ψ(x) ∈ K B), so ∀Z ⊇ Bn[(N∗, Z) |� ∀xψ(x)]. Therefore∀xψ(x) ∈ B1 (that is, to Bn+1, which is a subset of Bn+2).(b) n = l + 1. Then ∀x∀Z ⊇ Bl [(N∗, Z) |� (A2) ∧ (A3) → (N∗, Z) |� ψ(x)]. Ineffect, ∀Z ⊇ Bl [(N∗, Z) |� (A2)∧ (A3) → (N∗, Z) |� ∀xψ(x)] and this means that∀xψ(x) ∈ Bl+1 (that is, it belongs already to Bn).Case 4 αk is obtained in the proof from αi , α j , with i, j < k and α j = �αi → αk�.But then, by the inductive assumption, αi , α j ∈ Bn+2, so αk belongs to Bn+2 as well.

��The conclusion is that the Bn-sets provide natural models for all the Sn-s.

Corollary 21 ∀n(N∗, Bn) |� Sn.

Proof Fixn andψ ∈ Sn . Since Sn ⊆ Bn+1 (Fact 20),wehave:∀Z ⊇ Bn(if (N∗, Z) |�

(A2) ∧ (A3), then (N∗, Z) |� ψ). But (N∗, Bn) |� (A2) ∧ (A3) (Observation 19);

therefore (N∗, Bn) |� ψ . ��The next two observations can be established by induction on the length of the proofof an arbitrary ψ in BelCon(K ).

Fact 22 BelCon(K ) ⊆ Sω.

Fact 23 (N∗, Sω) |� BelCon(K ).

As a result, we obtain the corollary stating that the interior of BelCon(K ) is a subsetof Sω.

Corollary 24 I ntBelCon(K ) ⊆ Sω.

Proof Assume that BelCon(K ) � B(ψ) (in other words, we assume that ψ ∈I ntBelCon(K )). Then by Fact 23, (N∗, Sω) |� B(ψ), therefore ψ ∈ Sω. ��

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At this stage we are able to obtain a model for all sentences which are, provably inBelCon(K ), in the scope of the believability predicate.

Lemma 25 (N∗, Bω) |� I ntBelCon(K ).

Proof Since I ntBelCon(K ) ⊆ Sω (Corollary 24), it is enough to show that (N∗, Bω) |�Sω. Fixingψ ∈ Sω, we are going to prove that (N∗, Bω) |� ψ . By assumption, there isa natural number n such that ψ ∈ Sn . But Sn ⊆ Bn+1 (Fact 20), therefore ψ ∈ Bn+1.By definition of Bn+1, this means that:

∀Z ⊇ Bn[if (N∗, Z) |� (A2) ∧ (A3), then (N∗, Z) |� ψ].Since Bω ⊇ Bn and (N∗, Bω) |� (A2) ∧ (A3),38 we immediately conclude that(N∗, Bω) |� ψ . ��

Lemma 25 permits us to obtain Theorem 10 as a direct corollary. Given a model N∗of K , we can indeed expand it to the model (N∗, Bω), which satisfies I ntBelCon(K ).

Proceeding now to Theorem 11, we present below the proofs of two chosen cases.We are going to show that the believability of compositional axioms for negationand for the existential quantifier is derivable in Bel(T B−). Before we start, let usemphasise one technical point: if our base theory contains Peano arithmetic, we canapply gen also in cases with more than one initial universal quantifier. For example,given ‘∀xyB(ϕ(x, y))’ (with two quantifiers) as a theorem, we are allowed to conclude‘B(∀xy(ϕ(x, y)))’. The reason is that in PA (and therefore also in the scope of B)we can use the pairing function freely and so we can always get rid of additionalquantifiers.

Proof of Theorem 11 (chosen cases). In what follows we assume that quantifiers ofthe form ‘∀ψ’ and ‘∃ψ’ are restricted to arithmetical sentences (in effect, the readingis ‘for every (some) arithmetical sentence ψ’). For the compositional truth axiom fornegation, we claim that:

Bel(T B−) � B(∀ψ[T (¬ψ) ≡ ¬T (ψ)]).

The reasoning (carried out in Bel(T B−)) goes as follows:

(1) ∀ψ T B− � T (¬ψ) ≡ ¬T (ψ) (provable in PA)(2) ∀ψ B

(T (¬ψ) ≡ ¬T (ψ)

)(axiom (A1))

(3) B(∀ψ[T (¬ψ) ≡ ¬T (ψ)]). (by gen)

Analysing the case of the existential quantifier, we are going to show that:

Bel(T B−) � B(∀ϕ∀a(�∃aϕ� ∈ SentLPA → T (∃aϕ) ≡ ∃xT (ϕ(x)))

).

We start with presenting a proof of the following fact (with ‘Var ’ denoting the setof variables):

(*) Bel(T B−) � ∀ϕ(x)∀a ∈ Var B(∀a(T (ϕ(a)) ≡ ϕ(a))

).

The argument for (*) (carried out in Bel(T B−)) is given below. Let En(x) be a Σ1arithmetical predicate with the intuitive reading ‘x is a formula of syntactic complexitynot larger than n’. The expression ‘Trn’ abbreviates a partial truth predicate for En

formulas.

38 The second conjunct follows easily from Observation 19.

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(1) ∀ϕ(x)∀n∀x T B− � ϕ ∈ En → T (ϕ(x)) ≡ Trn(ϕ(x))(2) ∀ϕ(x)∀n∀x B

(ϕ ∈ En → T (ϕ(x)) ≡ Trn(ϕ(x))

)

(3) B(∀ϕ(x)∀n∀x [ϕ ∈ En → T (ϕ(x)) ≡ Trn(ϕ(x))])

(4) ∀ϕ(x)∀nB(∀x[ϕ ∈ En → T (ϕ(x)) ≡ Trn(ϕ(x))])(5) ∀ϕ(x)∀n[ϕ ∈ En → B

(∀x[T (ϕ(x)) ≡ Trn(ϕ(x))])](6) ∀ϕ(x)∀n(

ϕ ∈ En → T B− � ∀x[Trn(ϕ(x)) ≡ ϕ(x)])(7) ∀ϕ(x)∀n(

ϕ ∈ En → B(∀x[Trn(ϕ(x)) ≡ ϕ(x)]))(8) ∀ϕ(x)∀n(

ϕ ∈ En → B(∀x[T (ϕ(x)) ≡ ϕ(x)]))(9) ∀ϕ(x)B(∀x[T (ϕ(x)) ≡ ϕ(x)]))(10) ∀ϕ(x)∀a ∈ Var B(∀a[T (ϕ(a)) ≡ ϕ(a)]))

(1) is provable already in PA; (2) follows from (1) by axiom (A1); (3) is obtainedby gen. For (4), use the general statement ‘∀α(x)[B(∀xα(x)) → ∀x B(α(x))]’, whichis a theorem of Bel(T B−).39 (5) uses formalised Σ1-completeness: if ϕ ∈ Σn , then(since ‘ϕ ∈ Σn’ is Σ1) PA proves this fact, therefore B(ϕ ∈ Σn) and the conclusionfollows from (4). (6) is provable already in PA; (7) follows from (6) by (A1). Step(8) is obtained from (7) and (5). (9) follows from (8) together with the informationthat every formula is in En for some n; finally, (10) is obtained from (9) together with(A1).40

Given (*), we argue again in Bel(T B−) obtaining in the final move the composi-tional axiom for the existential quantifier.

(11) ∀ϕ(x)∀a ∈ Var B(∃aT (ϕ(a)) ≡ ∃aϕ(a)]))(12) ∀ϕ(x)∀a ∈ Var B(T (∃aϕ(a)) ≡ ∃aϕ(a)]))(13) ∀ϕ(x)∀a ∈ Var B(T (∃aϕ(a)) ≡ ∃aT (ϕ(a))]))(14) ∀ϕ(x)∀a ∈ Var B(T (∃aϕ(a)) ≡ ∃xT (ϕ(x))]))(15) B

(∀ϕ(x)∀a ∈ Var T (∃aϕ(a)) ≡ ∃xT (ϕ(x))]))

(11) follows from (*) by (A1) (the believability of logic); (12) holds because T B−is believable. (13) follows by (A1) from (11) and (12). Step (14) involves variablerenaming (a purely logical move); finally (15) is obtained by gen. ��

The proof of Theorem 15 uses the ideas which are very similar to those applied inthe proof of Theorem 11.

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